Prepivoting to reduce level error of confidence sets

TLDR: In this article, the root of the confidence set is transformed by its estimated bootstrap cumulative distribution function, and the transformation of a confidence set root by the estimated distribution function can be iterated one or more times with smaller error than do confidence sets based on the original root.

Abstract: SUMMARY Approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function's sampling distribution. We call this function the root of the confidence set. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. When the root is not a pivot, in the sense of classical statistics, the actual level of the approximate confidence set may differ substantially from the intended level. Prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function. Prepivoting can be iterated. Bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root. The first prepivoting is nearly equivalent to studentizing, when that operation is appropriate. Further iterations of prepivoting make higher order corrections automatically.